Math 137 Sample problems for Test I
1. The vector
a
starts from the point (1
,
1
,
1) and end at (2
,
2
,
2)
.
The vector
b
starts
from the point (1
,
2
,
3) and end at (2
,
3
,
4). Find
a
. Are vectors
a
and
b
diﬀerent?
2. Given
a
= (1
,
1
,
2) and
b
= (2
,
3
,
4), ﬁnd
a
·
b
and
a
×
b
.
3. The equations for two straight lines are
x
1
=
y
2
=
z
3
and
x
1
=
y
1
=
z

1
respectively. Do these two lines pass the point (0
,
0
,
0)? Are they perpendicular to each
other?
4. A plane contains three points: (0
,
0
,
0), (1
,
1
,
1) and (3
,
2
,
1).
(a) Find an equation for the plane.
(b) Find the area of the triangle with these three points as its vertices.
(c) Find a parametric equation for the line PERPENDICULAR TO THE PLANE
and passing through the point (1
,
2
,
3).
(d) Find the point P where the line in (c) passing through the plane.
(e) Find the distance between the point (1
,
2
,
3) to the plane, using the following two
methods: The ﬁrst one is to use the result in (c) and (d). The other is to use the formula